Response to dilatation and some soluble "squashing" models of interacting many-fermion systems

Abstract

The response of a many-fermion system to dilatation is studied through the introduction, in close analogy with the "cranking" and "pushing" models, of a "squashing" model characterized by the Hamiltonian, $H={H}_{0}\ensuremath{-}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\epsilon}}{D}_{x}$, where ${D}_{x}$ is the generator of dilatations in the $x$ direction. The evaluation of the "rigid" dilatational moment of such a noninteracting system is carried out for both standing wave and periodic boundary conditions in a cubic box of side $L$. As in the pushing model, the dilatation moment can be evaluated formally using the relation between ${D}_{x}$ and the commutator [${x}^{2}$, ${H}_{0}$]. For independent fermions moving in a harmonic oscillator one-body potential, this relation is shown to lead to further simplification after the introduction of perturbed one-body wave functions. In the case of periodic boundary conditions, it is shown that the rigid dilatational moment is not altered by an interparticle interaction to first order in the interaction strength; by comparison with the analogous cranking moment problem, this result extends to all orders in the interaction strength. The response to a symmetric dilatation, $D={D}_{x}+{D}_{y}+{D}_{z}$, is studied in a soluble Hartree model of "nuclear" correlations wherein a separable monopole-monopole interaction acts between "nucleons." In this model the dilatation moment in the presence of pair correlations is found to be the rigid moment, with the kinetic energy of dilatation $\ensuremath{\propto}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\epsilon}}}^{2}$, there being no higher order corrections.NUCLEAR STRUCTURE Approximate many-body methods applied to solvable models.